We investigate matrix algebras and tensor products of associative algebras over a commutative ring R with identity, such that the algebra satisfies a polynomial identity with coefficients in R. We call A a P. I. algebra over R if there exists a positive integer n and a polynomial f in n noncommuting variables with coefficients in R, not annihilating A, such that for all a 1 , … , a n {a_1}, \ldots ,{a_n} in A, f ( a 1 , … , a n ) = 0 f({a_1}, \ldots ,{a_n}) = 0 . We call A a P-algebra if f is homogeneous with at least one coefficient of 1. We define the docile identity, a polynomial identity generalizing commutativity, in that if A satisfies a docile identity, then for all n, A n {A_n} , the set of n-by-n matrices over A, satisfies a standard identity. We similarly define the unitary identity, which generalizes anticommutativity. Claudio Procesi and Lance Small recently proved that if A is a P. I. algebra over a field, then for all n, A n {A_n} satisfies some power of a standard identity. We generalize this result to P-algebras over commutative rings with identity. It follows that if A is a P-algebra, A satisfies a power of the docile identity.